V109_3 — A Local Scale Gap for Inverse Rational Zero‑Modes

Description: This paper proves unconditional local arithmetic results for inverse rational zero‑modes: an exact conductor‑drop identity, primitive odd‑conductor square‑root bounds, and weighted short‑modulus estimates. These establish that the completed inverse rational sum is oscillatory of square‑root scale on primitive support.

The global consequence is deliberately modest. On prime‑supported primitive ranges, the density term — given by the size of admissible sets relative to the prime modulus — remains bounded below by a positive constant, while the normalized oscillatory term has only square‑root size. Thus, under nonnegative weights supported away from small primes, density contributions and oscillatory zero‑mode contributions naturally live on different scales.

🔹 The Proved Local Core

• Exact conductor‑drop identity for inverse rational sums

• Primitive odd‑conductor square‑root estimate using Weil and p‑adic stationary phase

• Weighted short‑modulus bound showing oscillatory square‑root scale

🔹 Prime‑Supported Scale Gap

• Density contribution remains constant size relative to primes

• Oscillatory zero‑mode has only square‑root normalized size

• Proven mismatch: density and oscillation cannot match to lower order on prime‑supported ranges

🔹 Conditional Obstruction

• If weights are nonnegative and supported away from small primes, the density term dominates the oscillatory term.

• This mismatch is one‑sided: oscillation cannot approximate density to lower order.

🔹 Missing Mechanisms To achieve terminal closure, one of the following must be supplied:

• Support collapse (restricting effective support to collapsed conductors)

• Diagonal renormalization (density diagonal replaced by oscillatory baseline)

• Global defect cancellation (new theorem cancelling density versus oscillation mismatch)

🔹 Conclusion V109_3 establishes that local conductor collapse produces a clean scale gap between density and oscillation. The unconditional part is local: conductor drop, primitive bounds, and weighted short‑modulus control. The conditional part is global and abstract: density and oscillation functionals cannot match without an additional mechanism.